ABCDE is a pentagon. The angles $$A$$, $$B$$, $$C$$, $$D$$, $$E$$ are in the ratio $$8 \colon 9 \colon 12 \colon 15 \colon 10$$. The external bisector of B and the internal bisector of C meet at P. Then the measure of $$\angle BPC$$ is

The five angles of a pentagon add up to $$540^\circ$$ and the ratio has $$8 + 9 + 12 + 15 + 10 = 54$$ parts, so one part is $$10^\circ$$. This makes $$\angle B = 90^\circ$$ and $$\angle C = 120^\circ$$. The exterior angle at B is $$180^\circ - 90^\circ = 90^\circ$$, so its bisector makes $$45^\circ$$ with BC, and the internal bisector of $$\angle C$$ makes $$60^\circ$$ with CB. In the triangle BPC, $$\angle BPC = 180^\circ - 45^\circ - 60^\circ = 75^\circ$$.